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Spectral synthesis for exponentials and logarithmic length

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Abstract

We study hereditary completeness of systems of exponentials on an interval such that the corresponding generating function G is small outside of a lacunary sequence of intervals Ik. We show that, under some technical conditions, an exponential system is hereditarily complete if and only if the logarithmic length of the union of these intervals is infinite, i.e., \(\sum\nolimits_k {\int_{{I_k}} {{{dx} \over {1 + \left| x \right|}} = \infty } } \).

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Acknowledgement

The authors are grateful to the referee for numerous helpful remarks and suggestions.

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Authors and Affiliations

  1. Faculty of Mathematics and Mechanics, St. Petersburg State University, 28, Universitetskii prospekt, Staryi Petergof, 198504, Russia

    Anton Baranov

  2. Faculty of Mathematics and Computer Science, St. Petersburg State University, 29, Line 14th, Saint Petersburg, 199178, Russia

    Yurii Belov & Aleksei Kulikov

  3. Department of Mathematical Sciences, Norwegian University of Science and Technology, NO-7491, Trondheim, Norway

    Aleksei Kulikov

Authors
  1. Anton Baranov
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  2. Yurii Belov
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  3. Aleksei Kulikov
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Corresponding author

Correspondence to Anton Baranov.

Additional information

The results of Sections 2 and 3 were obtained with the support of Russian Science Foundation grant 19-71-30002. The results of Sections 4 and 5 were obtained with the support of Russian Foundation for Basic Research grant 20-51-14001-ANF-a.

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Baranov, A., Belov, Y. & Kulikov, A. Spectral synthesis for exponentials and logarithmic length. Isr. J. Math. 250, 403–427 (2022). https://doi.org/10.1007/s11856-022-2341-3

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  • DOI: https://doi.org/10.1007/s11856-022-2341-3

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